Random palindromes: multivariate generating function and Bernoulli density

Consider a nite alphabet with a probability distribution p. We study the probability (p) of obtaining a palindrome in a nite time by independent draws. Using a Mahler equation for an associated generating function, we give a closed form expression for (p). Moreover we describe completely the cases where (p) has value less than 1, in connection with the singularities of the generating function. Except for the case of a one or two letters alphabet it is found that (p) is always less than 1.